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Nonlinear Dissipative Oscillator withDisplaced Number StatesMartina Brisudová aa Institute of Physics, Slovac Academy of Sciences,Dúbravská Cesta 9, 84200, Bratislava, CzechoslovakiaVersion of record first published: 01 Mar 2007.

To cite this article: Martina Brisudová (1991): Nonlinear Dissipative Oscillator with DisplacedNumber States, Journal of Modern Optics, 38:12, 2505-2519

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JOURNAL OF MODERN OPTICS, 1991, VOL . 38, NO. 12, 2505-2520

Nonlinear dissipative oscillator with displaced numberstates

MARTINA BRISUDOVA

Institute of Physics, Slovac Academy of Sciences, Dubravska Cesta 9,

84200 Bratislava, Czechoslovakia

(Received 23 April 1991 ; revision received 14 June 1991)

Abstract. We study the interaction of a Kerr-like medium with light initiallyprepared in a displaced number state . We analyse squeezing properties andphoton statistics at the output of a Kerr-like medium. We show that undercertain conditions the superposition of two displaced number states can becreated . We study the influence of dissipation on the formation of the superposi-tion state .

1 . IntroductionThe nonlinear oscillator model [1-9] is a simple exactly solvable model

describing the optical Kerr effect . The behaviour of light in interaction with theKerr medium exhibits many interesting quantum features (for a review see[1, 2, 5, 8] and references therein) . In a number of papers the model was studiedfrom the point of view of the possible generation of macroscopically distinguishablesuperposition states [5,10-12] . It was shown by Yurke and Stoler [5] that the timeevolution of the anharmonic oscillator with an initial coherent state leads to aquantum superposition of two coherent states . A similar result was found forsqueezed states and a general expression for the complex amplitude of simultaneousposition and momentum measurement valid for an arbitrary initial state wasdetermined [10] .

Nevertheless, it is of crucial importance to examine the effect of externalfluctuations . Dissipation can be included in the model by coupling to either a zero-temperature [4, 11] or a non-zero-temperature [6,13,14] heat bath. The standardmethods are based on the master equation, which is valid for small nonlinearity andweak coupling. It was shown [4, 6,11,12] that the formation of the superpositionstate for both coherent- and squeezed-state inputs is very sensitive to dissipation .This fact can be explained, for example by the interpretation given in [13] . It isbased on splitting the antinormal quasi-distribution into quasi-distributions con-nected with discrete values of the intensity, which rotate with different angularvelocities. It was argued that the quantum coherence consists of the harmony of theorbits . The dissipation means that the system descends to lower values of intensitiesand particular quasi-distributions may rotate with velocities not preserving theharmony of motion. Milburn and Walls [12] have shown for squeezed states thatinterference arising from widely separated phase-space points is rapidly destroyed .

Recent developments in quantum optics enables us to generate a number stateof the electromagnetic field (see [15] and references therein) . Through the action of

0950-0340/91 $3 .00 © 1991 Taylor & Francis Ltd .

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a classical current on a cavity with a number state, a new state called the displacednumber state (DNS) can be derived. The various properties DNSs were studied byde Oliveira et al. [15] . Also they have shown that DNSs are much less sensitive todissipation than are the squeezed states .

Recently, the Kerr interaction has been investigated in detail with the displacedand squeezed state as an input [16] . Meanwhile, it was demonstrated for coherentstates that at times equal to fractions of the period the light evolves to asuperposition of the related number of the coherent states . The formations of thesesuperposition states were generalized as the higher-order (or fractional) revivals . Aminimal displacement which must be chosen in order to gain a macroscopicallydistinguishable superposition state via quantum revival of a given order was found .

We shall focus our main attention on the dissipative anharmonic oscillator withDNSs in the time of the second-order revival that is when the initial state evolvesinto a superposition of two DNSs, as we shall show . States of this kind exhibitinteresting quantum features and are of current interest (V . Buzek 1991, privatecommunication) . For this purpose, we shall use DNSs with a relatively largedisplacement. A comparison between DNSs and coherent states with similarintensities will be given .

The paper is organized as follows . First, we review some properties of theDNSs. The anharmonic oscillator is presented and we solve the master equation .We find the mean values of operators necessary for evaluation of squeezing andstatistical properties . Then we study the effect of dissipation on superposition stateformation by means of the Q function . The results are summarized in section 4, theconclusions .

2 . Displaced number statesIn this section we shall briefly review some properties of DNSs . Detailed

discussion can be found in [15] .A DNS can be defined as a result of the action of a displacement operator D(a)

on a number state I1> [15, 17-19] :

Let us note that for 1=0 the DNS reduces to the well known coherent stateintroduced by Glauber .

In what follows we shall use the number state representation12, 1>=Y C`) nln>,

(3)n

where- z

C«) = expIaI

~(-a*)t-n0!)n~ 1Jz Lrt - "(IaI 2 ),

l>,n,rt

( 2

(4)

expZalz)an-1( `!)1/zLi-`(Ialz),

1,<n,

where Li- '(Ial 2 ) are associated Laguerre polynomials [20] :

la, l>=D(a)Il>, (1)where the displacement operator has the standard form

D(a) = exp (aa+ - a*a) . (2)

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Nonlinear dissipative oscillator

~

m

L;_1(x)=

(-1)mn x

M=O

(I-M m l.

Because IC(,l) I2 includes a polynomial in n of degree one, one can eventually find 1zeros in the photon number distribution. An interpretation of oscillations in photonnumber distribution can be found in [15] .

Perhaps one of the clearest ways to study various effects in phase space is via theQ function . It can be defined as [2, 14, 21]

Q(/1)=«IpIQ>,

(5)where I#> is a coherent state and p the system density matrix . For a pure state kG>the definition can be written as

Q(P)= 10 I0l 2 ,

(6)The Q function gives the phase-space probability density of simultaneous measure-ment of dimensionless position and momentum .

Q(/3) for a DNS can be expressed in terms of the Q function for the appropriatenumber state [15], namely

Q.,,(#)=Qi(fl-a)=exp(-I/i-aI2) Ia1!

al"

(7)

As is seen from the Q function (figure 1), the state becomes phase dependent andis centred around a new origin located at the coherent-field amplitude position .

2507

Figure 1 . The Q functions for the DNS lot= 20 1 / 2 , 1=1>. It has the form of the appropriatenumber state Q function located at the coherent-state-amplitude position .

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In order to evaluate the effect of imperfect measurement, the influence ofdissipation on the photon distribution was studied by de Oliveira et al. [15] . Theyproved that DNS quantum features are relatively robust in comparison withsqueezed states .

3. Anharmonic oscillator modelThe Hamiltonian of the nonlinear oscillator coupled to a reservoir of zero-

temperature oscillators is [1]

H=wa+ a+xa+2a2 + Y (g;abi++g, a+ b;),

(8),

where a and a + are the photon annihilation and creation operators, w is thefrequency of light, x is the nonlinear coupling, b1 and b;+ are the boson annihilationand creation operators and g; is the coupling constant of the interaction with thereservoir .

Standard treatments [22] lead to a master equation for the reduced systemdensity operator in the interaction picture (a-*exp (i (o t) a) :

dP = _ ix[a+2a2, P] +2 [ap, a+ ] + 2 [a, pa +],

(9)

where y is the damping constant .A formal solution can be written as [23]

p(t) = T (t)p(0),

(10)

where T (t) is defined by

00

T (t) = m Nm(t)p(0),m=o

with Nm(t) given by

where

fr

im

t2dtm

dtm_i . . .0

0

f 0Nm(t) =

dtl S(t - tm)JS(tm - tm_ i) . . . JS(tl),

JP=Yapa + ,

S(t)P=exp(axta +2a2 -Za + at)pexp ixta+2a2 -2a+at

NO(t) =_S(t) .

We consider the system at t = 0 being in a DNS . Thus p(O) can be expressed as :

P(0) =

C(~)Cm")In><mI,n, m

with C,~,`) given by equation (4) .Following the same treatment as in [11], namely substituting p(O) in the form

(11) into equation (10) and calculating T (t) jn><mj, the solution of the masterequation is

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Nonlinear dissipative oscillator

1(ln-m)2 min (n,m)

n!m!

1/2 1

n,m

2

k=p {(n-k)!(m-k)!) k!

x {µ(T) exp [-i(n-m)T]}("+m-2k)Kµ(T)

+i(exp i(n

m)T]})kIn-kX<m-kI],n-m

J)

where T=2Xt is the scaled time, K=y/2x and p(T)=exp(-KT) .The operator solution p(T) enables all moments to be determined as

<a+P(T)a9(T)> = Tr [P(T)a+Pa9] .

(13)

For the purpose of consequent analysis we determine

<a(T)>=[p(T)]1/2 E (n+1)112C,(,

l+1Cn

j)

+l~l(T)exp(-1T) "-1'

(14)n =o

K+i

<a2(T)>=µ (T) exP (-iT)

L(n+2)(n+l)1/zCnl+2C;,(~)(K+2iµ(T)exp(-i2T)\" (15)

n=o

K+2i

<a+(T)a(T)> =µ(T)(l+ IaI 2 ),

(16)

<[a+(T)a(T)]2>=µ2(T)(l+IaI2))2+p2(T)l(2IaI2-1)+P(T)(l+IaI2) .

(17)

In particular, for 1=1, equations (14) and (15) can be easily expressed as aproduct of a polynomial and an exponential function, giving an opportunity toperform analytical study.

3 .1 . Light squeezingKnowing <a+ a>, <a2> and <a> it is easy to evaluate squeezing properties via S;

functions (i= 1, 2 corresponding to the first and the second quadrature) measuringthe deviation from the corresponding uncertainty of a coherent state (see [2, 8] andreferences therein) :

51 =2<a +a>+2 Re <a2>-4(Re <a>)2 ,

(18)

S2 =2<a+ a>-2 Re <a 2>-4(Im <a>)2 .

(19)

Light is squeezed if S; < 0 and the maximal squeezing is obtained if S; _ -1 .In figures 2 (a) and (b) we plot functions S 1 and S2 against one period of the

scaled time T for an initial state Ia = 20 1 /2 , 1=1 > without dissipation .In figure 3 we plot S 1 (T) against IaI 2 at the scaled time T=n for 1=1 and K=0,

when S 1 reaches its minima for any IaI 2 . Despite the fact that for coherent states onecan obtain squeezing [8], even one incoherent photon (i .e . the photon related to thenumber state 11 > of the DNS D(a)I1 > which is present in a field even in the limit ofthe coherent amplitude Ial to zero [15]) ensures that no squeezing occurs in the caseof the DNS .

On a short time scale, during the first moments, S2 tends to rise while S 1decreases. For IaI 2 > 1, the local extremas of S 1 occur at times proportional to 1 /IaI 2 .

We plot the first and for strong fields the global maximum of S 1 against Ia1 2 in figure4. There is no squeezing .

2509

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M. Brisudovd

80.00 -

S 1

60.00 -

40.00 -

20.00 -

0.00 10.00

3.14

6.28T

(a)

100.00 -

S2

80.00 -

60.00 -

40.00 -

20.00 -

I0.00

0.00

3.14

6.28T

(b)

Figure 2. The S. functions for the initial state Ia=20 1 "Z, 1=1> against one period of thescaled time. (a) Initially S l tends to decrease. Another minimum can be found at 2=n.(b) SZ initially increases . The squeezing occurs in neither of the quadratures .

1

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0.60

S1(T=7T)

0.50 -

0.40 -

0.30 -

0.20 -

0.10

0.00

,

,

,

,

10.25

0.30

0.35

0.40

0.45

0.5IaI 2

Figure 3 . The S l function at T=n when it reaches its minimum for any lal 2 against Joel' .There is no squeezing .

0.40 -

S1

a l(T= I(XI2t )

0.30 -

0.20 -

0.10 -

Nonlinear dissipative oscillator

2511

0.00 0

-5o

1000

I a I215(

Figure 4 . The S l function at i=const ./lal2 when it is minimized on a short time scale .There is no squeezing .

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2512

M. Brisudovd

(a)

(b)Figure 5 . The Q function at i=it for the initial state ja=1, 1=1> : (a) without dissipation,

K=O; (b) K=0'1 .

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Nonlinear dissipative oscillator

2513

3 .2 . Photon statisticsIn order to evaluate the statistics of the light we determine the Mandel [24] Q

parameter :

Q= <(Da +a)2 >-<a+a>

(20)

<a+a>

For Poissonian statistics, Q = 0 . If Q<0, the light is said to be sub-Poissonian ;otherwise it is super-Poissonian .

Using equations (16) and (17) one obtains

z -Q=µ(T)1 21+ ~ a1 21 ,

(21)

which is equal to the Mandel parameter of the initial DNS times µ(T) [15] . The statehas sub-Poissonian statistics if IaI 2 < 1 . From the above, it is apparent thatdissipation does not change this property up to T-+oo, when it reduces to thevacuum .

3 .3 . The superposition state formationPrevious workers have shown that an initial state, because of interaction with a

Kerr medium modelled as an anharmonic oscillator, evolves into a superposition oftwo quantum states at a certain time [5, 10] . Such behaviour can be distinguishedfor DNS, too . Indeed, at the scaled time T=n the light is described by the state

00

W

jO(T=71)>= E (-1)'CZI,I2j>+ F (- 1)'Czj+1 12j+ 1 > .

This state is a superposition of two DNSs :

IIG(T = TC) = 2 1/z[exp (_i4+i2l)lia,l>+exp(i_il)l 4l-ia,l >] .

(23)

Substituting equation (23) into the definition of the Q function (6), after somearrangement, one obtains

Qa.I(N, T =71)_ IQI(f + la) + JQI(fl - la)

+(-1)I[QI(f+ia)QI(f-ia)]112sin[l(F +F 2)+2Im(iafl*)],

(24)

with l 1 and F 2 given by

cos T1 + i sin F1= (IIfi-iaI

II+iacos r2 + i sin T 2= IP + iad

The factor + i and - i corresponding to the rotation through T=n/2 with respect toresults in [5,10-12] is caused by the difference in the Hamiltonians considered ; in

particular we use a +2a2 instead of (a +a)Z .Including dissipation, Q(f, T) can be calculated with the aid of the density

matrix (12) and it reads for an arbitrary time

(22)

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2514 M. Brisudovl

(b)

(a)

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Nonlinear dissipative oscillator

2515

lP

(c)Figure 6 . The Q function at z=n for the initial state la=201)2 , 1=0>: (a) K=0 ;

(b) K=0'01 ; (C) K=0. 1 .

6

,

ao

\ min(m,n)

[n)m)] 1/2

1Q(f,t)=exp( - If1 2 ) n,Y o C,(,l)Cn; l) exp (n-m)2'

k=0 [(n-k)!(m-k)!k!

(n 1 +m-2k) K{1-µ(z ) exp [-i(n-m)z]} k.(n-k) m-kx {µ(z) exp [-i(n-m)t]}

2

(

K+i(n-m)

)

]

(25)

For an illustration, we plot Q($, z = it) for the initial state Ia =1, l=1(figure 5(a)) . Since the interference term in equation (24) is comparable withQ1(fl+i) or Q1($-i), the superposition is `hidden' under interference fringes .

Figure 5 (b) shows the same state but for K=0 . 1 (i .e . µ(r)=0.73). The picturehas become smoother, but the two valleys are still recognizable .

In figure 6 (a), one can clearly see a superposition state which has beengenerated from the initial coherent state la=20 1/2 , 1=0> . Figure 6(b) shows thateven a very small dissipation K=0 .01 (i .e. µ(z)=0 .97) prevents the formation of thesuperposition state . The two peaks can be recognized, but their heights havedecreased by roughly 1 . 7 times. We can also see the formation of a ring-like shape .

Under the influence of dissipation K=0 .1 (figure 6(c)) the form of Q(#) hasdrastically changed . Q(fl) has become centrally symmetric and the height hasdecreased approximately six times .

Figure 7 shows the superposition states generated from an initial DNS with l=1and figure 8 from an initial DNS with 1 =2 and 1=5 . Comparing figures 6. (a) with7(a), one can easily distinguish whether there was any incoherent photon in theinitial state . The presence of incoherent photons manifests itself as a zero value ofthe Q function at the coherent-amplitude position .

The effect of incoherent photons is relatively robust with respect to dissipation(see figure 7 (b) and 8 (a) and (b) with K=0 . 1). Dissipation causes not only a loss ofphotons but also a deterioration of phase information . The state tends to becomecentrally symmetric . Nevertheless, since there were minima at coherent-amplitudepositions, one can discern a saddle point in the dissipative Q function . Further-more, there are two hills on the imaginary axis originating from the maxima locatednearest to the centre .

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251 6 M. Brisudovk

(a)

(b)

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Nonlinear dissipative oscillator

2517

(c)Figure 7. The Q function at T=n for the initial state Ia=20 112 , 1=1> : (a) K=0 ;

(b) x=0 .1 ; (c) K=0 .5 .

For the initial state ja=201 / 2 , 1=1>, the form of the Q function (figure 7(b))differs slightly from that of the coherent state Ia= 20 1 / 2> under the same dissipationK = 0 . 1 (figure 6 (c)) . The difference is more apparent for the initial states with l= 2and 1=5, namely fa=20 1 '2 , 1=2> (figure 8(a)) and Ia=17 112 , l=5> (figure 8(b)) .Figure 7 (c) demonstrates that, for the case of the initial state with 1=1, for largeenough dissipation x=0 . 5 (µ(T)=0 .2) this interesting structure disappearscompletely.

4 . ConclusionsWe have investigated the dissipative anharmonic oscillator with DNSs . We have

evaluated the Mandel Q parameter . The dissipation does not change the statisticsuntil the state reduces to the vacuum .

Concerning squeezing properties for the non-dissipative case, we have foundthat a DNS as an input of an anharmonic oscillator does not become squeezed . Ithas already been argued [19] that, while the phase uncertainty of a DNS is greaterthan that of a coherent state, the degree of squeezing in the case of the DNS must beconsiderably smaller. We have shown that even one incoherent photon suppressessqueezing completely.

Then we have examined the superposition state formation at r=i. For the non-dissipative case, we have shown that the Q(fl) function at this time can be split intoQ(f) functions corresponding to two DNSs and an interference term .

For the purpose of obtaining the macroscopically distinguishable superpositionstate we have focused on states with a relatively large displacement . Includingdissipation, the formation of the superposition state is strongly suppressed .

The dissipation smoothes the shape of Q(fl) and tends to make the state centrallysymmetric or, in other words, the phase information evaporates . A further feature

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2518

M. Brisudovk

-7_

0

(a)

(b)

Figure 8. The Q function at T=it and K=0 . 1 for the initial state : (a) Ia=20' 12 , 1=2) ;

(b) lot = 17 1 / 2 , 1= 5> .

becomes more apparent for greater I f l . The saddle point in Q(/3) originates from theminima of a non-dissipative Q function . Meanwhile, one can recognize two hillslocated on the imaginary axis, which are the rest of the non-dissipative Q functionmaxima. In this way, the information on incoherent photons in the initial state ispreserved .

The superposition DNS seem to be convenient for studying the effect ofdissipation from the point of view of its dependence on phase-space position .Comparing fields with almost equal intensities (n = 22 ; see figure 8) under the samedissipation, one can see that the structure described above is more distinct the

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Nonlinear dissipative oscillator

2519

greater the number component of the field, that is the smaller the displacement .Furthermore, one can see that the most resistant are those structures of Q functionwhich are located nearest to the centre. Thus our result supports the conclusion ofMilburn and Walls [12] .

AcknowledgementsIt is a pleasure for me to acknowledge a fruitful discussion with V . Buiek. I am

also indebted to P. Striienec for his help with plotting the figures .

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Reidel) .[2] TANA§, R., 1984, Coherence and Quantum Optics, Vol . 5, edited by L. Mandel and

E. Wolf (New York: Plenum), p . 643 .[3] MILBURN, G. J ., 1986, Phys . Rev . A, 33, 674 .[4] MILBURN, G. J ., and HOLMES, C., 1986, Phys . Rev . Lett ., 56, 2237 .[5] YURKE, B ., and STOLER, D., 1986, Phys. Rev. Lett ., 57, 13 .[6] PEiiINOVA, V., and LuKS, A., 1988, J . mod. Optics, 35, 1513 .[7] PEtINovA, V., LuKI§, A., and KARSKA, M., 1990, J. mod. Optics, 37, 1055 .[8] BU EK, V., 1989, Phys. Lett . A, 136, 188 ; TANA9, R., 1989, Phys . Lett . A, 141, 217 .[9] Bu~EK, V., 1989, Phys. Rev . A, 39, 5432 .

[10] TOMBESI, P., and MECOZZI, A., 1987, J. opt . Soc. Am . B, 4, 1700 .[11] MILBURN, G. J ., MECOZZI, A., and ToMBESI, P., J . mod. Optics, 36, 1607 .[12] MILBURN, G. J ., and WALLS, D. F ., 1988, Phys. Rev . A, 38, 1087 .[13] PEhINovA, V., and LUK§, A., 1990, Phys . Rev . A, 41, 414 .[14] DANIEL, D . J ., and MILBURN, G. J ., 1989, Phys. Rev . A, 39, 4628 .[15] DE OLIVERIA, F. A. M ., KIM, M. S ., KNIGHT, P . L., and BU EK, V., 1990, Phys. Rev . A,

41, 2645 .[16] KRAL, P., 1990, Phys . Rev. A, 42, 4177 .[17] SATYANARAYA, V., 1985, Phys . Rev . D, 32, 400 .[18] Roy, S . M., and SINGH, V., 1982, Phys . Rev . D, 25, 3413 .[19] BUIEK, V., JEx, I ., and BRISUDovA, M., 1991, Int . J. mod. Phys . B, 5, 797 .[20] ABRAMOWITZ, M ., and STEGUN, I . A., 1972, Handbook of Mathematical Functions (New

York: Dover Publications) .[21] MILBURN, G. J ., 1988, Squeezed and Nonclassical Light, edited by P . Tombesi and

E. R. Pike (New York : Plenum) .[22] LOUISELL, W. H., 1933, Quantum Statistical Properties of Radiation (New York: Wiley) .[23] DAVIES, E . B., 1973, Quantum Theory of Open Systems (New York: Academic Press) .[25] MANDEL, L., Optics Lett ., 4, 205 .

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